Lyapunov-type inequalities for the modified discrete Helmholtz operator on $\mathbb{Z}$

We establish Lyapunov-type inequalities for Dirichlet boundary value problems driven by the modified discrete Helmholtz operator on finite balls of the integer lattice $\mathbb{Z}$. This yields weighted inequalities for scalar problems and a spectral-radius criterion for coupled systems, as well as sharpness results. As an application, we study a weighted eigenvalue problem and obtain explicit two-sided bounds for the first eigenvalue by combining a Lyapunov-type lower estimate with a variational upper bound. Numerical illustrations are provided for localized weights.